System and method for controlling superconducting quantum circuits using single flux quantum logic circuits

ABSTRACT

A system and methods for controlling superconducting quantum circuits are provided. The system includes at least one superconducting quantum circuit described by multiple quantum states, and at least one single flux quantum (“SFQ”) control circuit configured to generate a voltage pulse sequence that includes a plurality of voltage pulses temporally separated by a pulse-to-pulse spacing timed to a resonance period. The system also includes at least one coupling between the at least one superconducting quantum circuit and the at least one SFQ control circuit configured to transmit the voltage pulse sequence generated using the SFQ control circuit to the at least one superconducting quantum circuit. In some aspects, the system further includes a controller system configured to optimize the pulse-to-pulse spacing to minimize a gate infidelity due to at least one of a timing error, a timing jitter and a weak qubit anharmonicity.

BACKGROUND

The field of the disclosure is related to superconducting circuits. Moreparticularly, the disclosure relates to systems and methods forcontrolling superconducting quantum circuits using single flux quantum(“SFQ”) logic.

In the field of quantum computation, the performance of quantum bits(“qubits”) has advanced rapidly in recent years, with preliminarymulti-qubit implementations leading toward surface code architectures.In contrast to classical computational methods that rely on binary datastored in the form of definite on/off states, or bits, methods inquantum computation take advantage of the quantum mechanical nature ofquantum systems. Specifically, quantum systems are described using aprobabilistic approach, whereby a system includes quantized energylevels whose state may be represented using a superposition of multiplequantum-mechanical states.

Among several implementations currently being pursued,superconductor-based circuits present good candidates for theconstruction of qubits given the low dissipation inherent tosuperconducting materials, which in principle can produce coherencetimes necessary for performing useful quantum computations. In addition,because complex superconducting circuits can be micro-fabricated usingconventional integrated-circuit processing techniques, scaling to alarge number of qubits is relatively straightforward. In particular,superconducting circuits that include Josephson tunnel junctions,generally composed of two superconducting electrodes separated by a thininsulator, may be utilized for scalable quantum information processingin the solid state. Such Josephson junction-based superconductingcircuits are advantageous on account of their strongly nonlinearbehavior, which allows a breaking of degeneracy for the transitionfrequencies, and thus restricting system dynamics to specific quantumstates.

Presently, gate and measurement fidelities are within reach of thethreshold for fault-tolerant quantum computing based on topologicalsurface codes, and hence there is interest in scaling quantum computingdevices that include a few qubits to much larger, multi-qubit circuitry.However, a superconducting quantum computer that will outperform thebest available classical machines may necessitate thousands if notmillions of physical qubits, and hence the wiring architecture andcontrol of a such large-scale quantum processor presents a formidabletechnical challenge.

Present systems for measurement and control of superconducting quantumcircuits typically include low-temperature systems, such as dilutionrefrigeration units. Such systems are configured with microwavefrequency generators and single-sideband mixing hardware that generateand transmit microwave electromagnetic signals to multiplesuperconducting circuits for purposes of measurement and control of thestate of each qubit. However, such systems are limited in terms ofwiring availability, as well as thermal and noise coupling to roomtemperature electronics. Hence, in applications involving cryogenictemperatures it is highly desirable to integrate as much of the controland measurement circuitry for a multi-qubit system as possible into inorder to reduce wiring heat load, latency, power consumption, and theoverall system footprint.

Given the above, there a need for systems and methods yielding scalablequantum computation that includes the ability to perform rapidhigh-fidelity control and measurement of both single qubits andmulti-qubit parity, while controlling the resources utilized.

SUMMARY

The present disclosure overcomes the aforementioned drawbacks byproviding a system and methods for controlling superconducting quantumcircuits. Specifically, the present disclosure describes an approach foruse in coherent manipulation of harmonic oscillator and qubit modesusing a sequence of voltage pulses generated using single flux quantum(“SFQ”) circuits. Particularly, the present disclosure utilizes coherentrotations obtained by using a pulse-to-pulse spacing timed to a periodof a target oscillator. Advantageously, control by way of voltage pulsesgenerated using SFQ circuits coupled to multiple superconducting quantumsystems may be achieved in a low-temperature cryostat without need forapplying microwave electromagnetic signals. As will be described,calculation of gate errors due to timing jitter of SFQ-derived voltagepulses and due to weak anharmonicity of a qubit demonstrate gatefidelities in excess of 99.9 percent are achievable for sequence lengthsof, for example, roughly 20 nanoseconds.

In one aspect of the present disclosure, a quantum computing system isprovided. The system includes at least one superconducting quantumcircuit described by multiple quantum states, and at least one singleflux quantum (“SFQ”) control circuit configured to generate a voltagepulse sequence that includes a plurality of voltage pulses temporallyseparated by a pulse-to-pulse spacing timed to a resonance period. Thesystem also includes at least one coupling between the at least onesuperconducting quantum circuit and the at least one SFQ control circuitconfigured to transmit the voltage pulse sequence generated using theSFQ control circuit to the at least one superconducting quantum circuit.

In another aspect of the present disclosure, a method for controllingsuperconducting quantum circuits is provided. The method includesproviding at least one superconducting quantum circuit described bymultiple quantum states and coupled to at least one single flux quantum(“SFQ”) control circuit, and generating, using the at least one SFQcontrol circuit, a voltage pulse sequence that includes a plurality ofvoltage pulses temporally separated by a pulse-to-pulse spacing. Themethod also includes applying the voltage pulse sequence to the at leastone superconducting quantum circuit to achieve an excitation consistentwith a target transition between the multiple quantum states.

In yet another aspect of the present disclosure, a method forcontrolling superconducting circuits is provided. The method includesproviding at least one superconducting quantum circuit described bymultiple quantum states and coupled to at least one single flux quantum(“SFQ”) circuit, and generating, using the at least one SFQ controlcircuit, a voltage pulse sequence that includes a plurality of voltagepulses temporally separated by a pulse-to-pulse spacing timed to aresonance period. The method also includes applying the voltage pulsesequence to the at least one superconducting quantum circuit to achievean excitation consistent with a target transition between the multiplequantum states.

The foregoing and other aspects and advantages of the invention willappear from the following description. In the description, reference ismade to the accompanying drawings which form a part hereof, and in whichthere is shown by way of illustration a preferred embodiment of theinvention. Such embodiment does not necessarily represent the full scopeof the invention, however, and reference is made therefore to the claimsand herein for interpreting the scope of the invention.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic of a quantum computing system in accordance withthe present disclosure.

FIG. 2 is a flowchart setting forth steps of one example of a processfor controlling superconducting quantum circuits in accordance with thepresent disclosure.

FIG. 3a is a schematic of an example single flux quantum (“SFQ”) pulsesequence for achieving an excitation for a resonator coupled via acapacitance to a SFQ-based voltage source.

FIG. 3b shows a trajectory in quadrature space for a cavity driven by aresonant SFQ pulse sequence, accordance with the present disclosure.

FIG. 3c shows a trajectory on the Bloch sphere for a qubit driven with aresonant SFQ pulse sequence, in accordance with the present disclosure.

FIG. 4a is a graphical illustration of polar θ and azimuthal φ angles ofthe Bloch vector following an SFQ-based (π/2)_(y) rotation for differentrealizations of timing jitter.

FIG. 4b is a graphical illustration of SFQ gate error versus jitter foran SFQ (π/2)_(y) rotation.

FIG. 5 is a graphical illustration of the average gate error and |2>state error for SFQ pulse trains versus number of pulses n.

FIG. 6 is a graphical illustration showing the dependence of gate erroron number pulses n used to realize a (π/2)_(y) rotation for SFQ pulsesof finite width.

FIGS. 7a and 7b provide a schematic illustration of SFQ pulse timingjitter for internal and external clocking modes.

FIG. 8 is a graphical illustration showing the dependence of gate errordue to pulse timing jitter on rotation angle θ for SFQ pulses generatedby an external clock (top panel) and an internal clock (bottom panel).

FIG. 9 is a graphical illustration showing the dependence of gate erroron qubit anharmonicity for (π/2)_(y) (solid line) and π_(y)(dash-double-dot line) rotations.

DETAILED DESCRIPTION

Previously, single flux quantum (“SFQ”) superconducting circuits havebeen investigated as a high-speed and ultra low-power digital logicalternative to current CMOS-based integrated circuits. Specifically, atthe heart of SFQ circuits are Josephson junctions, whose fast switchingcapabilities (on the order of a few picoseconds), provide a significantimprovement over conventional approaches. The operating principle of theSFQ circuits is based on single 2π-leaps in the phase φ of overdampedJosephson junctions. Due to large damping, driven Josephson junctionsnever switch completely into the phase-running regime having a largeaverage voltage across the junction, but instead generate short voltagepulses whose time integral equals the superconducting flux quantumΦ₀=h/2e. Generally, SFQ digital logic circuits generate, manipulate andstore classical bits of information, or logical “0” and “1” values,using voltage pulses, or fluxons, that propagate ballistically alongpassive superconducting microstrip lines or active Josephsontransmission lines.

There have been some experimental demonstrations implementing SFQ-basedcircuits for qubit biasing, along with fluxon-based schemes for qubitmeasurement. In addition, there has been a proposal to generatemicrowave pulses for qubit control using appropriately filtered SFQpulse trains, although the required filter and matching sections wouldbe challenging to realize practically. In spite of such work, there hasbeen no compelling realization thus far of coherent control of quantumsystems using direct excitation of system modes via SFQ-generatedvoltage pulses. This is because each single SFQ pulse produces abroadband excitation, which is not directly useful for coherentmanipulation of quantum circuits, since it does not offer thepossibility for providing selective excitation of individual transitionsbetween quantum states.

By contrast, the present disclosure recognizes that superconductingquantum systems, such as qubits or resonator cavities, may be coherentlycontrolled using voltage pulses generated using such SFQ circuits, aswill be described. Specifically, systems and methods that implementresonant SFQ pulse sequences are provided for the coherent control ofquantum system modes, such as qubit and resonator cavity modes. Examplesof superconducting qubits include charge qubits, flux qubits, phasequbits, transmons, xmons, and so forth. As will be described, SFQ-basedgates, in accordance with the present disclosure, are robust againstleakage errors and timing jitter of the applied voltage pulses, withhigh achievable fidelities in nanosecond gate times.

Turning to FIG. 1, an example for a quantum computing system 100, inaccordance with the present disclosure is shown. The quantum computingsystem 100 includes superconducting quantum circuit(s) 102, which mayinclude a single or multiple superconducting qubits or resonatorcavities, coupled to any single flux quantum (“SFQ”) control circuit(s)104, via one or more couplings 106. The superconducting quantumcircuit(s) 102 may include any number of linear and nonlinear circuitelements, including Josephson junctions, inductors, capacitors,resistors, and so on. The quantum computing system 100 is configured tooperate over a broad range of temperatures, including temperaturesconsistent with a superconducting state for materials configuredtherein.

In accordance with embodiments of the present disclosure, the SFQcontrol circuit(s) 104, which may include a number SFQ modules arrangedin any manner, may be configured to generate a voltage pulse sequence,or pulse train. Specifically, the voltage pulse sequence may include anynumber of periodic or non-periodic voltage pulses, temporally separatedby pulse-to-pulse spacings, for use in attaining target transitionsbetween multiple quantum states describing the superconducting quantumcircuits(s) 102. In some aspects, the pulse-to-pulse spacings may betimed to one or more resonance periods such that excitations consistentwith target transitions between the quantum states of thesuperconducting quantum circuits(s) 102 are achieved. This may beaccomplished via controller system(s) 108, which can include anyelectronic systems, hardware or circuitry components in communicationwith the SFQ control circuit(s) 104, and configured to initiate andcontrol the timing and intensity of voltage pulses generated therefrom.For example, such controller system(s) 108 may include capabilities forproviding signals of any amplitude, frequency, waveform, or duration, asinput signals to the SFQ control circuit(s) 104, in the form of current,voltage, magnetic flux signals, and so forth. In some aspects, thecontroller system(s) 108 may be further configured to optimize thepulse-to-pulse sequence in order to minimize a gate infidelity due toany combination of timing errors, timing jitter and weak qubitanharmonicity, along with other errors.

The one or more couplings 106 providing a communication between thesuperconducting quantum circuit(s) 102 and SFQ control circuit(s) 104may configured to transmit, modulate, amplify, or filter, the voltagepulse sequence generated using the SFQ control circuit(s) 104. Suchcouplings 106 may include any circuitry elements, including capacitiveor inductive elements, passive superconducting microstrip lines, activeJosephson transmission lines, including any number of Josephsonjunctions, and so forth. In some aspects, the controller system(s) 108may further provide signals for modulating or tuning the one or morecouplings 106.

In certain desired configurations, the one or more couplings 106 may bedesigned such that nonequilibrium quasiparticles generated in the SFQcontrol circuit(s) 104 are isolated from the superconducting quantumcircuit(s) 102 in a manner intended to avoid the introduction of degreesof freedom leading to quantum decoherence. For example, quasiparticlepoisoning can be mitigated by avoiding direct galvanic connectionbetween the signal and ground traces of the SFQ control circuit(s) 104and superconducting quantum circuit(s) 102.

In some configurations of the quantum computing system 100, a readoutsystem(s) 110 in communication with the superconducting quantumcircuit(s) 102 may also be included, whereby the readout system(s) 110may be configured to provide readout information in relation tocontrolled quantum states of the superconducting quantum circuit(s) 102,in accordance with the present disclosure.

Turning to FIG. 2, a flowchart is shown setting forth steps of a process200 for controlling superconducting quantum circuits with SFQ logiccircuits, in accordance with the present disclosure. The process beginsat process block 202 where any superconducting quantum circuit(s) andSFQ control circuit(s) are provided. As described, such superconductingquantum circuit(s) may include single or multi-qubit quantum systems orresonant cavities. At process block 204, a voltage pulse sequence, orpulse train, may be generated using the SFQ control circuits, viacontroller systems, hardware and circuit elements. The voltage pulsesequence may include any number of voltage pulses temporally separatedby pulse-to-pulse spacings. In some aspects, a pulse-to-pulse spacingmay be timed to one or more resonance periods of a superconductingquantum system in order to generate excitations consistent with targettransitions between quantum states of the system. In addition, thepulse-to-pulse sequence may be optimized in order to minimize a gateinfidelity due to any combination of timing errors, timing jitter andweak qubit anharmonicity, along with other errors.

At process block 206, the generated voltage pulse sequence is thenapplied to the superconducting quantum circuit(s) via at least onecoupling to achieve target transitions between states of the quantumsystem(s). Then at process block 208 a report may be generated of anyshape or form. Specifically, information with respect to states of thequantum system (s) may be obtained via signals from single or multiplereadouts by way of readout system(s), as mentioned.

In previous approaches, control of superconducting qubits is generallyaccomplished using shaped microwave pulses that realize arbitraryrotations over the Bloch sphere. Amplitude modulation of a resonantcarrier wave concentrates drive power at the frequency of interest, andthe microwave pulses are shaped to minimize power at nearby transitionfrequencies to avoid excitation out of the qubit manifold. As described,such approaches are limited in their applicability to scalable quantumcomputing systems as would inherently necessitate complex and expensiveresources.

By contrast, in accordance with systems and methods of the presentdisclosure, intuition for the effectiveness of an arbitrary drive pulsein achieving a desired transition (or avoiding an undesired one) betweenquantum states, can be obtained by considering a simple classical modelof an LC resonator. Specifically, a drive waveform is coupled to theresonator from a time-dependent voltage source V(t) through a couplingcapacitance C_(c) is considered, as shown in FIG. 3a . The energydeposited in such a resonator is given by:

$\begin{matrix}{E = {\frac{\omega_{0}^{2}c_{c}^{2}}{2C^{\prime\;}}{{\overset{\sim}{V}\left( \omega_{0} \right)}}^{2}}} & (1)\end{matrix}$

where C′=C+C_(C), ω₀=1/√{square root over (LC′)} and where the tilderepresents the Fourier transform {tilde over (V)}(ω)=∫_(−∞)^(∞)V(t)e^(jωt)dt. The energy coupled to the resonator is proportionalto the energy spectral destiny of the drive waveform at the resonatorfrequency.

Herein, the response of a microwave resonator to an SFQ pulse is ofinterest. For state-of-the-art Nb-based SFQ technology, characteristicpulse amplitudes are roughly 2 mV and pulse widths around 1 picosecond.As the pulse widths are much less than the period of the microwaveresonator, an SFQ voltage pulse can be modeled as a Dirac δ-function V(t)=Φ₀δ(t). In this case, {tilde over (V)}(ω)=Φ₀ and Eqn. (1) reducesto:

$\begin{matrix}{E_{1} = \frac{\omega_{0}^{2}C_{c}^{2}\Phi_{0}^{2}}{2C^{\prime}}} & (2)\end{matrix}$

where the subscript 1 indicates reference to the response to a singlepulse. Because the SFQ pulse width is much smaller than the oscillatorperiod, the energy deposited is quite insensitive to the detailed shapeof the SFQ pulse, and is determined rather by the time integral of thepulse, which is precisely quantized to a single flux quantum. Forexample, for a Gaussian SFQ pulse with standard deviation τ, the aboveresult is modified by the pre-factor e^(−ω) ⁰ ² ^(τ) ² , which yields acorrection of 0.02% for τ=0.5 ps and ω₀/2π=5 GHz

As mentioned, a single SFQ pulse produces a broadband excitation. Forthis reason, a single pulse is not useful for coherent manipulation ofquantum circuits, since it does not offer the possibility to selectivelyexcite individual transitions. The picture changes, however, whenconsidering driving the resonator with a sequence of SFQ voltage pulses.Particularly, to coherently excite a resonator a pulse-to-pulseseparation that is timed to the resonator period may be used. Suchapproach is analogous to pumping up a swing by giving a short push onceper cycle, in contrast to sinusoidally forcing the swing throughout anentire period of oscillation. A driving voltage may then be consideredas:V _(n)(t)=Φ₀[δ(t)+δ(t−T)+ . . . +δ(t−(n−1)T)]  (3)

where T is the separation between pulses and n is the number of pulses.It may be found that the pulse sequence couples an energy to theresonator equal to:

$\begin{matrix}{E_{n} = {\frac{\omega_{0}^{2}C_{0}^{2}}{2C^{\prime\;}}\frac{\sin^{2}\left( {n\;\omega_{0}{T/2}} \right)}{\sin^{2}\left( {\omega_{0}{T/2}} \right)}}} & (4)\end{matrix}$

By way of example, it is worthwhile to consider the energy transferredby an SFQ pulse sequence to a typical cavity mode in a superconductingcQED circuit. By taking ω₀/2π=5 GHz, C=1 pF, and C_(c)=1 fF, it isdetermined that a single SFQ pulse couples roughly 6×10⁻⁴ quanta to thecavity mode. By contrast, for a resonant pulse sequence, in accordancewith the present disclosure, with T equal to an integer multiple ofcavity periods, the pulses add coherently, so that the total energydeposited in the cavity goes as n². Because of such quadratic scaling,about 40 pulses are required to populate the cavity with a singleexcitation, accomplished in a time of roughly 40·2π/ω₀=8 ns.

A recent proposal for cQED measurement based on microwave countingrelies on the preparation of “bright” and “dark” cavity pointer statesusing a coherent drive pulse with length matched to the inverse detuningof the dressed cavity frequencies. This protocol is readily adapted to aSFQ excitation of the readout cavity. For a qubit-cavity system withdressed cavity resonances at ω₀−χ or (ω₀+χ) corresponding to the qubit|0

(or |1

) states, an SFQ pulse train with interval T=2/π(ω₀+χ) and total numberof pulses n=(ω₀+χ)/2χ will coherently populate the cavity if the qubitis in the |1

state, while returning the cavity to the vacuum upon completion of thesequence if the qubit is in the |0

state.

By way of example, the response of the quantum oscillator to SFQexcitation is considered. The time-dependent circuit Hamiltonian iswritten as:

$\begin{matrix}{H = {\frac{\left\lbrack {\hat{Q} - {C_{C}{V(t)}}} \right\rbrack^{2}}{2C^{\prime}} + \frac{{\hat{\Phi}}^{2}}{2L}}} & (5)\end{matrix}$

The Hamiltonian may be decomposed into the unperturbed free HamiltonianH_(free) and a time-dependent excitation Hamiltonian H_(SFQ):

$\begin{matrix}{{H_{free} = {\frac{{\hat{Q}}^{2}}{2C^{\prime}} + \frac{{\hat{\Phi}}^{2}}{2L}}}{H_{SFQ} = {{- \frac{C_{C}}{C^{\prime}}}{V(t)}\hat{Q}}}} & (6)\end{matrix}$

In terms of the usual raising and lowering operators,

$\begin{matrix}{{H_{free} = {\hslash\;\omega_{0}{\hat{a}}^{\dagger}\hat{a}}},{H_{SFQ} = {{\mathbb{i}}\; C_{C}{V(t)}\sqrt{\frac{\hslash\;\omega_{0}}{2C^{\prime}}}{\left( {\hat{a} - {\hat{a}}^{\dagger}} \right).}}}} & (7)\end{matrix}$

The effect of the SFQ pulse is then to induce a coherent displacement ofthe cavity state by amount

$\begin{matrix}{\alpha_{SFQ} = {C_{C}\Phi_{0}\sqrt{\frac{\omega_{0}}{2\hslash\; C^{\prime\;}}}}} & (8)\end{matrix}$

as shown in FIG. 3b . The energy deposited by the pulse matches theclassical expression (2). A sequence of n pulses produces a coherentstate with amplitude

$\alpha_{n} = {\alpha_{SFQ}{\sum\limits_{k = 0}^{n - 1}\;{\exp\left( {{- {\mathbb{i}}}\; k\;\omega_{0}T} \right)}}}$and mean energy E_(n)=

₀|α_(n)|² consistent with the classical expression (4).

Next, by way of another example, the application of SFQ pulses may beconsidered for a two-level qubit. The Hamiltonian of the system becomes

$\begin{matrix}{{H_{free} = {\frac{{\hslash\omega}_{10}}{2}\left( {I - {\hat{\sigma}}_{z}} \right)}}{{H_{SFQ} = {C_{C}{V(t)}\sqrt{\frac{{\hslash\omega}_{10}}{2C}}{\hat{\sigma}}_{y}}},}} & (9)\end{matrix}$

where I is the identity matrix and {circumflex over (σ)}_(y,z) are theusual Pauli matrices. In the limit of a short, intense SFQ pulse, adiscrete rotation of the state vector about they-axis by angle δθ isinduced, namely

$\begin{matrix}{{\delta\theta} = {C_{C}\Phi_{0}\sqrt{\frac{2\omega_{10}}{\hslash\; C}}}} & (10)\end{matrix}$

In between pulses, the qubit evolves under the influence of H_(free).The SFQ pulse sequence may then induce coherent rotations when the freeevolution periods are timed to the oscillation period 2π/ω₁₀ of thequbit, as shown in FIG. 3c . For a qubit initially in state |0

, the resonant pulse sequence induces a coherent rotation in thexz-plane. For a pulse interval that is slightly mismatched from theoscillation period, the state vector slowly drifts away from thexz-plane, and in the limit of a large timing mismatch the state vectorundergoes small excursions about the north pole of the Bloch sphere.

As may be appreciated from Eqn. 10, the angle of rotation induced by theSFQ pulse depends on the strength of the capacitive coupling to thequbit, which may be taken as fixed. While tunable inductive couplershave been demonstrated, it is unclear that they could be engineered toperform well on the picosecond timescales characteristic of an SFQpulse. For that reason, it might prove advantageous to work with a fixedrotation angle once the coupling to the qubit is determined by thecircuit design. For small rotation angle δθ˜0.01, the resulting gateerror is at most δθ²/4. Moreover, this error can be further reduced byappropriately tailoring the timing delay between the SFQ pulses.

Other potential sources of errors in SFQ-based gates include timingjitter of the pulses and weak anharmonicity of the qubit. By way ofexample, in the following, the six eigenstates of the Pauli operatorsare taken as input states, and gate error is computed as the state erroraveraged over these input states; this approach is similar tointerleaved random benchmarking with single-qubit Clifford gates.

The effect of a timing error δt the SFQ pulse is to induce a spuriousrotation of the state vector by angle ω₁₀ δt sin θ, where θ is theinstantaneous polar angle of the state vector. It may be assumed thatthe arrival times of the individual pules are distributed normally withstandard deviation σ. To consider the effect of timing jitter onrotations derived from SFQ pulse trains, the manner in which the SFQcircuit is clocked may need to be specified. Specifically, if the pulsetrain is derived from a stable external frequency source (used, forexample, to trigger a DC/SFQ converter), the timing jitter per pulse isindependent of the length of the pulse train. Timing errors associatedwith each pulse may be largely compensated by the following pulse, anderror in the final pulse dominates error in the sequence as a whole.Pulse timing jitter leads to the average gate error

$\begin{matrix}{{1 - F_{avg}^{ext}} = {\frac{\left( {\omega_{10}\sigma} \right)^{2}}{6}\left\lbrack {\frac{\Theta^{2}}{n} + 1} \right\rbrack}} & (11)\end{matrix}$

where the superscript “ext” refers to the mode of clocking the SFQ pulsetrain from a stable source. For practical purposes this timing jitterwill introduce negligible gate error.

Next, a more demanding case where pulse timing errors accumulaterandomly is considered, so that the timing jitter in the nth pulse is√{square root over (n)} larger than the timing jitter in the initialpulse. This could be the situation, for example, when the SFQ pulsetrain is generated internally from an SFQ clock ring. In this case, thedeviation of the state vector from the desired trajectory grows as√{square root over (n)} leading to a degradation of gate fidelity thatscales linearly with n. The timing jitter results in an average gateerror

$\begin{matrix}{{{1 - F_{avg}^{int}} = \frac{{n\left( {\omega_{10}\sigma} \right)}^{2}}{6}},} & (12)\end{matrix}$

where the superscript “int” refers to the internal clock used togenerate the pulse train.

In the thermal regime, the timing jitter of the SFQ pulse scales as thesquare root of temperature, and average timing jitter per junction of0.2 ps has been measured in a large-scale SFQ circuit operated at 4.2 K.For an SFQ circuit operated at reduced temperature in a dilutionrefrigerator, the timing jitter is expected to be lower, althoughquantum fluctuations may lead to non-negligible jitter even for circuitsoperated at millikelvin temperatures. Moreover, if the SFQ pulse sourceis coupled to the qubit sample via a long Josephson transmission lineconsisting of N junctions, a qubit may see a √{square root over (N)}degradation of the timing jitter due to the sequential switching of thejunctions in the line.

By way of example, Monte Carlo simulations of gate error due to timingjitter for an SFQ (π/2)_(y) rotation realized from 100 pulses wereperformed, in the case where timing errors of the pulse generatoraccumulate incoherently, cf. Eqn. (12). The results are shown in FIG.4a-b . Specifically, for the |0

state input, timing errors lead predominantly to y-errors and result inan average infidelity given by Eqn. (33), as will be described. Smallz-errors accumulate coherently and lead to a systematic underrotation ofthe state vector. For the input (|0

+i|1

)/√{square root over (2)} which ideally is unaffected by the (π/2)_(y)rotation, timing errors initially provide kicks in the x-direction; oncex-errors are allowed to accumulate, subsequent SFQ pulses generateadditional z-errors. In FIG. 4b the average gate error versus pulsetiming jitter σ is shown. For σ=0.2 ps, the average gate error is6.6×10⁴.

A practical superconducting qubit is not an ideal two-level system. Fora typical transmon qubit, the anharmonicity (ω₁₀−ω₂₁)/φ₁₀ is of order4-5%. A single strong SFQ pulse may induce a large spurious populationof the |2

state as a result of its broad band-width, and leakage errors induced byfast SFQ control pulses have been considered previously. However, aresonant SFQ pulse train tailored to perform a desired rotation in the0-1 subspace in a larger number of steps n, in accordance with thepresent disclosure, may show greatly reduced spectral density at ω₂₁enabling high-fidelity SFQ-based gates with acceptable leakage. Considera three-level system with unperturbed Hamiltonian

$\begin{matrix}{H_{free} = \begin{pmatrix}0 & 0 & 0 \\0 & {\hslash\omega}_{10} & 0 \\0 & 0 & {\hslash\left( {\omega_{10} + \omega_{21}} \right)}\end{pmatrix}} & (13)\end{matrix}$

The charge induced on the qubit capacitance by the SFQ pulse leads tothe Hamiltonian

$\begin{matrix}{H_{SFQ} = {{\mathbb{i}}\; C_{C}{V(t)}\sqrt{\frac{{\hslash\omega}_{10}}{2C}}\begin{pmatrix}0 & {- 1} & 0 \\1 & 0 & {- \sqrt{2}} \\0 & \sqrt{2} & 0\end{pmatrix}}} & (14)\end{matrix}$

Here, typical transmon parameters ω₁₀/2π=5 GHz and ω₂₁/2π=4.8 GHz areconsidered. Gate fidelity and |2

state errors for resonant SFQ pulse trains designed to produce (π/2)_(y)and π_(y) rotations for a range of total numbers of pulses (and hencegate durations) are examined. In addition, results for the |2

state leakage

for the (π/2)_(y) gate for qubit states |j

=|0

,|1

are shown in FIG. 5. Gate error is dominated by leakage to the |2

state. Gate errors decrease as n⁻² by increasing the number of pulsesand thus the total duration of the sequence, as one reduces the spectralweight of the pulse sequence at the 1-2 transition. Moreover, gate errorexhibits an oscillatory behavior, with minima corresponding to pointswhere there is destructive interference at the leakage transition. Forthe (π/2)_(y) pulse, fidelity of 99.9% is achieved in 100 pulses,corresponding to a 20 ns gate time for a 5 GHz qubit, while for a πpulse 99.9% fidelity is achieved in around 300 pulses.

In analyzing error in qubit rotations realized from resonant trains ofsingle flux quantum (SFQ) pulses, three sources of error are considered:(1) finite width and (2) timing jitter of the SFQ pulses, and (3)leakage of the qubit wave function to the second excited state. Foraccessible SFQ pulse parameters, gate errors due to all three sourcesare shown to be suppressed well below 10⁻³.

Specifically, the ideal SFQ-based gate, where coherent rotations arerealized from 5 function pulses with no timing error and where the qubitis treated as an ideal two-level system, is compared to the actualSFQ-based gate, where the pulses have finite width and timing jitter andwhere weak anharmonicity of the qubit is explicitly taken into account.The state-averaged overlap fidelity of a qubit gate is computed asfollows:

$\begin{matrix}{{F_{avg}\left( {{??}_{id},{??}_{G}} \right)} = \frac{2 + {{{Tr}\left( {{??}_{id}^{\dagger}{??}_{G}} \right)}}^{2}}{6}} & (15)\end{matrix}$

where

_(id) is the unitary time evolution operator for the ideal gate andU_(G) corresponds to the actual gate. To evaluate fidelity of SFQ-basedrotations by angle θ about they-axis, let

$\begin{matrix}{{??}_{id} = {\exp\left( \frac{{\mathbb{i}\Theta}{\hat{\sigma}}_{y}}{2} \right)}} & (16)\end{matrix}$

This rotation may be composed from n smaller rotations by angle δθ=θ/nabout the y-axis, interspersed with appropriate free precessionintervals that are matched to the Larmor period 2π/ω₁₀ of the qubit. Theunitary operator describing the δ-function pulses is given as follows:

$\begin{matrix}{U_{\delta}^{(1)} = {\exp\left( \frac{{\mathbb{i}\delta\theta}{\hat{\sigma}}_{y}}{2} \right)}} & (17)\end{matrix}$

Similarly, free precession for interval t is described by the unitaryoperator

$\begin{matrix}{{U_{f}(t)} = {\exp\left( \frac{{\mathbb{i}\omega}_{10}t\;{\hat{\sigma}}_{z}}{2} \right)}} & (18)\end{matrix}$

The actual evolution operator U_(G) may be composed of a product ofsingle-pulse evolution operators

_(G) ⁽¹⁾ and free evolutions between pulses. It may be assumed that theSFQ pulse vanishes outside the time interval (−t_(c),t_(c)) and that theevolution during the pulse is defined by the differential equation

$\begin{matrix}{{{\mathbb{i}\hslash}\frac{\partial{U_{G}^{(1)}(t)}}{\partial t}} = {{H(t)}{U_{G}^{(1)}(t)}}} & (19)\end{matrix}$

with the initial condition U_(G) ⁽¹⁾(−t_(c))=1. The evolution operatorat time t_(c) thus defines the overall effect of a single pulse on thequbit state:

_(G) ⁽¹⁾(δθ)=U_(G) ⁽¹⁾(−t_(c))

In the following, the structure of the actual evolution operatorsU_(G)(θ) and gate infidelity for three sources of error are considered.

Finite Pulse Width

To consider the effect of the finite SFQ pulse width, begin byconsidering rectangular SFQ pulses with width 2t_(c). The fullHamiltonian during the pulse is

$\begin{matrix}{H = {{\frac{{\hslash\omega}_{10}}{2}\left( {I - {\hat{\sigma}}_{z}} \right)} - {\frac{\hslash\delta\theta}{4t_{C}}{\hat{\sigma}}_{y}}}} & (20)\end{matrix}$

where δθ is the rotation angle induced by a single pulse. Thecorresponding evolution operator during the pulse represents precessionin the field (0, δθ/2t_(C), ω₁₀) and has the form,

_(rect) ⁽¹⁾=exp(i(2ω₁₀ t _(C){circumflex over (σ)}_(z)+δθ{circumflexover (σ)}_(y))/2).  (21)

In the δ-function approximation, the evolution during the same timeinterval would be

_(id) ⁽¹⁾=exp(iω ₁₀ t _(C){circumflex over (σ)}_(z)/2)

_(δ) ⁽¹⁾exp(iω ₁₀ t _(C){circumflex over (σ)}_(z)/2)  (22)

Using Eqn. 15, the overlap error 1−F_(1, rect) for a single pulse up tofourth order in t_(c) and δθ is obtained:

$\begin{matrix}{{1 - F_{1,{rect}}} = {\frac{1}{216}\left( {{{\delta\theta}^{4}\omega_{10}^{2}t_{C}^{2}} + {{\delta\theta}^{2}\omega_{10}^{4}t_{C}^{4}} - \frac{{\delta\theta}^{4}\omega_{10}^{4}t_{C}^{4}}{5}} \right)}} & (23)\end{matrix}$

This expression gives the important message that for short pulses, tothe lowest order in ω₁₀t_(C), the error decreases as δθ⁴ for decreasingδθ However, for very small δθ

ω₁₀t_(C) the error becomes quadratic in the rotation angle δθ. The gateerror due to rectangular pulses is shown in FIG. 6 as a dash-dottedtrace.

Next, a SFQ pulse may be modeled by a Gaussian shape of width τ:

$\begin{matrix}{{V(t)} = {\frac{\Phi_{0}}{\sqrt{2{\pi\tau}}}{\mathbb{e}}^{{- t^{2}}\text{/}2\tau^{2}}}} & (24)\end{matrix}$

The time-dependent Hamiltonian is given by:

$\begin{matrix}{{H(t)} = {{\frac{h\;\omega_{10}}{2}\left( {I - {\hat{\sigma}}_{z}} \right)} + {\frac{\hslash\delta\theta}{2\sqrt{2{\pi\tau}}}{\mathbb{e}}^{{{- {({t - t_{k}})}^{2}}/2}\tau^{2}}{\hat{\sigma}}_{y}}}} & (25)\end{matrix}$

where t_(k) is the arrival time of the SFQ pulse. The time evolutionoperator for the full Gaussian pulse is denoted

_(Gauss) ⁽¹⁾, and this operator is computed over the interval (−t_(C),t_(C)), where t_(C)=5τ. It is assumed that the SFQ pulse vanishesoutside of the time interval and that qubit evolution is described bythe free evolution operator U_(f) (2π/ω₁₀−2t_(C)) during the time2π/ω₁₀−2t_(C). The gate error for a single Gaussian SFQ pulse can beevaluated according to Eqn. 15 and the result is also shown in FIG. 6 asa dotted trace. The error for the Gaussian pulse closely follows theresult for rectangular pulses with proper choice of τ.

In addition, the fidelity of a gate consisting of a resonant train of nGaussian SFQ pulses that is designed to realize a rotation by angleθ=nδθ about the y axis is analyzed. The gate evolution operator iswritten as

_(G)(θ)=[U _(f)(2π/ω₁₀ −t _(C))

_(Gauss) ⁽¹⁾(δθ)U _(f)(−t _(C))]^(n)  (26)

Substituting this expression to Eqn. 15, the gate fidelity can beobtained. In FIG. 6, the gate error as a function of the number ofpulses n for θ=π/2 and for Gaussian pulses with width τ=4 ps is shown.The average gate error for a full rotation is observed to be n² timeslarger than the error of a single pulse, 1−F_(avg)=n²(1−F₁). For largervalues of n, the single pulse error scales as 1/n² [cf. Eqn. 23] and theaverage gate error 1−F_(avg)∝(ω₁₀τ)⁴θ² becomes independent of n, whileremaining below 10⁴ due to the factor (ω₁₀τ)⁴. Note that in thesesimulations very long pulse times are assumed compared to what isachieved in practical SFQ circuits (where pulse widths τ<1 ps arereadily accessible) in order to circumvent numerical errors associatedwith finite machine precision; the scaling of gate error with pulseduration can be understood from Eqn. 23. For practical SFQ pulses, errorassociated with finite pulse duration may be much smaller than the othertwo errors analyzed below.

Pulse Timing Jitter

Small variation in the arrival times of the SFQ pulses presents anothersource of gate error. As described, the effect of timing jitter on SFQgate fidelity depends on the manner in which the SFQ timing generator istriggered. Hence, the following two cases are considered: (1) Externalclock. Here, the SFQ pulses are derived from a stable external clock, sothat the timing error per pulse does not grow with the length of thesequence. (2) Internal clock. Here, there is fixed error in thepulse-to-pulse spacing, so that errors in the timing of individualpulses accumulate incoherently as the length of the sequence grows. Theeffect of these two different clocking modes on pulse timing jitter isdepicted schematically in FIGS. 7a and 7b . These two cases are examinedin detail below. For further discussion, an alternative expression toevaluate gate fidelity is utilized:

F avg = 1 6 ⁢ ∑ α ⁢ F α , ⁢ F a = Tr ⁢ { G ⁢ ρ α ⁢ G † ⁢ id ⁢ ρ α ⁢ id † } ( 27 )

where the average is performed over the Pauli eigenstates ρ_(α)=|α

α| aligned along directions α=±x, ±y, ±z.

External Clock

The effect of timing jitter on pulse trains derived from a stableexternal clock is analyzed. It is assumed that the pulse arrival timesare distributed normally with respect to the external clock withdistribution with σ. For small jitter, ω₁₀σ

1, F_(α) is evaluated using the following analysis. The evolution of aqubit is characterized by a sequence of discrete rotations, Eqn. 17,interspersed with intervals of free precession that are nominallymatched to the qubit period 2π/ω₁₀. Due to pulse timing jitter, theactual free precession interval between the (k−1)th and kth pulsesbecomes 2π/ω₁₀+δt_(k)−δt_(k-1), where δt_(k) is the timing errorassociated with the kth pulse. For a qubit state vector that isinitially aligned along the z-axis, the timing error causes the state toacquire a component δy_(k) in the y-direction:δy _(k)=ω₁₀(δt _(k) −δt _(k-1))sin(kδθ)  (29)

Here kδθ is the instantaneous polar angle of the qubit state vector.During the gate operation, the qubit state vector accumulates the errorδY=Σ_(k)δy_(k), finding:

$\begin{matrix}{{F_{z\;} = {1 - {\overset{\_}{\delta\; Y^{2}}/4}}}{{\overset{\_}{\delta\; Y^{2}} = {\left( {\omega_{10}\sigma} \right)^{2}\left\lbrack {{\sin^{2}\left( {n\;\delta\;\theta} \right)} + {{\delta\theta}^{2}{\sum\limits_{k = 1}^{n - 1}{\cos^{2}\left( {k\;{\delta\theta}} \right)}}}} \right\rbrack}},}} & (30)\end{matrix}$

where the overbar represents an average over pulse jitter times δt_(k).Assuming that δθ=θ/n is small, the summation may be replaced byintegration in the last expression and finding

$\begin{matrix}{F_{x} = {1 - {\left( {\omega_{10}\sigma} \right)^{2}\left\lbrack {{\frac{\Theta^{2}}{8n}\left( {1 + \frac{\sin\; 2\Theta}{2\Theta}} \right)} + \frac{\sin^{2}\Theta}{4}} \right\rbrack}}} & (31)\end{matrix}$

for a qubit state initially aligned along the z-direction.

For a qubit state initially aligned along the x-axis, the analysis isthe same with the replacement of sin(kδθ) with cos(kδθ) in Eqn. 29. Inthis case

$\begin{matrix}{F_{x} = {1 - {\left\lbrack {{\frac{\Theta^{2}}{8n}\left( {1 - \frac{\sin\; 2\Theta}{2\Theta}} \right)} + \frac{\cos^{2}\Theta}{4}} \right\rbrack\left( {\omega_{10}\sigma} \right)^{2}}}} & (32)\end{matrix}$

In the above expressions for F_(x) and F_(z), a small error along thez-direction is disregarded, which is higher order in ω₁₀σ.

In the case of a qubit state vector initially aligned along the y-axis,the state vector remains close to they-axis, and after each freeprecession acquires an error in the x-directionδy_(k)≅ω₁₀(δt_(k)−δt_(k-1)). This error is then rotated by the remainingn−k pulses in xz plane, resulting in the accumulation of total gateerror along the x- and z-directions δX=Σ_(k)δt_(k) cos(Θ−kδθ) andδZ=Σ_(k)δt_(k) sin(Θ−kδθ). For a qubit state initially aligned alongthey-axis, the gate fidelity is

$\begin{matrix}{F_{y} = {1 - \frac{\overset{\_}{\delta\; X^{2}}}{4} - \frac{\overset{\_}{\delta\; Z^{2}}}{4}}} & (33)\end{matrix}$

Evaluating the summations for δX and δZ under the assumption ofuncorrelated δt_(k), the following is obtained

$\begin{matrix}{F_{y} = {1 - {\frac{\left( {\omega_{10}\sigma} \right)^{2}}{4}\left( {\frac{\Theta^{2}}{n} + 1} \right)}}} & (34)\end{matrix}$

The average gate error is computed from Eqn. 27, yielding

$\begin{matrix}{F_{avg} = {1 - {\frac{\left( {\omega_{10}\sigma} \right)^{2}}{6}{\left( {\frac{\Theta^{2}}{n} + 1} \right).}}}} & (35)\end{matrix}$Internal Clock

Next, gate fidelity is evaluated for a system where the SFQ pulses areclocked internally in such a way that the time interval between pulsesfluctuates independently, so that error in the arrival times ofindividual pulses accumulates incoherently. The free evolution isdetermined by the time interval 2π/ω₁₀+δt_(k), where δt_(k) is normallydistributed and uncorrelated from pulse to pulse. Due to the timingerror, a qubit state vector initially aligned along the z-directionacquires a spurious component δt_(k)=δt_(k) sin(kδθ) along the y-axis.Thus δY² =(ω₁₀σ)²Σ_(k) sin²(kδθ). Following the same procedure describedin the previous section, the gate fidelity is found to be

$\begin{matrix}{F_{z} = {1 - {{\frac{{n\left( {\omega_{10}\sigma} \right)}^{2}}{8}\left\lbrack {1 - \frac{\sin\; 2\Theta}{2\Theta}} \right\rbrack}.}}} & (36)\end{matrix}$

For a pure state initially aligned along the x-axis,

$\begin{matrix}{F_{x} = {1 - {{\frac{{n\left( {\omega_{10}\sigma} \right)}^{2}}{8}\left\lbrack {1 + \frac{\sin\; 2\Theta}{2\Theta}} \right\rbrack}.}}} & (37)\end{matrix}$

For states initially aligned along the y-axis, error accumulates alongboth the x- and z-directions, as discussed in the previous section.Evaluating the corresponding gate errors δX² and δZ² , the followingexpression is found

$\begin{matrix}{F_{y} = {{1 - \frac{\overset{\_}{X^{2}}}{4} - \frac{\overset{\_}{Z^{2}}}{4}} \simeq {1 - {\frac{{n\left( {\omega_{10}\sigma} \right)}^{2}}{4}.}}}} & (38)\end{matrix}$

The gate fidelity averaged over all qubit states is then given by

$\begin{matrix}{F_{avg} = {1 - \frac{{n\left( {\omega_{10}\sigma} \right)}^{2}}{6}}} & (39)\end{matrix}$

The SFQ gate error in the presence of timing jitter as a function ofrotation angle θ for pure initial states aligned along directions α={x,y, z} was numerically evaluated, taking σ=0.2 ps and n=100. For a givenrealization of timing jitter {δt_(k)}, the overlap of the final qubitstate was calculated with the corresponding state obtained by the idealgate, Eqn. 16, and then the overlap was averaged over 10⁴ realizationsof {δt_(k)}. The results are shown in the upper and lower panels of FIG.8 for external and internal gating of the SFQ pulses, respectively. Thesimulation results are plotted as points, and the lines represent theanalytical expressions derived above.

Higher Energy Levels of the Qubit

Finally, the effect of weak qubit anharmonicity on SFQ gate fidelity isanalyzed. The qubit is treated as a three-level system withanharmonicity η=(ω₁₀−ω₂₁)/ω₁₀. The Hamiltonian is given by Eqns. 13 and14. The corresponding time evolution operator is a three-dimensionalunitary matrix and the definition for the average fidelity has to bemodified accordingly. However, given the interest in averaging over thetwo-level qubit subspace of the system Hilbert space, the averagefidelity reduces to

F avg ⁡ ( id ⁢ , G ) = Tr ⁢ { G † ⁢ P ⁢ G ⁢ P } +  Tr ⁢ { P ⁢ id † ⁢ G }  2 6 (40 )

where P is the projection operator on the qubit subspace. Thisexpression for fidelity is consistent with Eqn. 27 provided thefollowing modified three-dimensional unitary operator is used todescribe evolution under the ideal gate:

$\begin{matrix}{U_{id} = \begin{pmatrix}{\cos\left( {\Theta/2} \right)} & {\sin\left( {\Theta/2} \right)} & 0 \\{- {\sin\left( {\Theta/2} \right)}} & {\cos\left( {\Theta/2} \right)} & 0 \\0 & 0 & 1\end{pmatrix}} & (41)\end{matrix}$

The error of a θ_(y) gate due to the presence of the second excitedstate is evaluated by summing the spurious amplitude of the |2

state induced by pulse k as

where

is the probability amplitude of the qubit being in the first excitedstate at time of pulse k if it was initially in state |j

with j=0,1. Here the factor exp(2πiη(n−k)) represents the phase acquiredby the second excited state over the remainder of the sequence followingthe kth pulse. Performing summation over n SFQ pulses, the probabilityof excitation to the second excited state is obtained as:

$\begin{matrix}{p_{2}^{j\rangle} = {\frac{\Theta^{2}}{8n^{2}}{{\frac{1 - {\mathbb{e}}^{{\mathbb{i}}\;{n{({{2{\pi\eta}} + {{\delta\theta}/2}})}}}}{1 - {\mathbb{e}}^{{\mathbb{i}}{({{2{\pi\eta}} + {{\delta\theta}/2}})}}} - {\left( {- 1} \right)^{j}\frac{1 - {\mathbb{e}}^{{\mathbb{i}}\;{n{({{2{\pi\eta}} - {{\delta\theta}/2}})}}}}{1 - {\mathbb{e}}^{{\mathbb{i}}{({2{{\pi\eta\delta\theta}/2}})}}}}}}^{2}}} & (42)\end{matrix}$

Here it is assumed that the |1

state amplitudes

=sin (kδθ/2) and

cos(kδθ/2) are not significantly modified by the small amount of leakageto the second excited state, and direct |0

→|2

transitions are disregarded.

Numerical analysis of the average fidelity F_(avg) of (π/2)_(y) andπ_(y) gates in the case of weak qubit anharmonicity is shown in FIG. 5,where F_(avg) vs. number of SFQ pulses n for a qubit with spectrumω₁₀/2π=5 GHz and ω₂₁/2π=4.8 GHz is shown. For the (π/2)_(y) gate, theprobabilities P₂ of qubit excitation to the |2

state for the qubit initially in the ground (dotted line) or the firstexcited state (dashed line) are also plotted. It is noteworthy that thenumerically evaluated curves for P₂ are well described by Eqn. 42 for n

10. As mentioned, the fidelity decreases as n⁻² for large n, in additionto displaying an oscillating component that is more pronounced forsmaller gate rotation angle θ.

In FIG. 9 the average gate error as a function of anharmonicity η for(π/2)_(y) and π_(y) gates realized using n=100 SFQ pulses is shown. Theinfidelity drastically decreases for n

1/n and then exhibits a slower decrease with a minimum at η=½. Theoscillations of 1−F_(avg) have the period Δη

1/n and nearly disappear for a π_(y) rotation. In FIG. 9, the |2

state occupation

following the (π/2)_(y) rotation for the qubit initially in state |j

is also shown. A comparison of numerical calculations of

with Eqn. 42 shows that the two agree well for |η|

1/n.

Simulations, as described herein, indicate that errors due to weak qubitanharmonicity and SFQ timing jitter are roughly of the same order ˜10⁻³for SFQ pulse trains consisting of around 100 pulses, corresponding to20 ns for a π/2 rotation of a 5 GHz qubit. While these errors arenon-negligible, they are nevertheless small enough to enable robustqubit control with fast gates at error levels below the threshold for afault-tolerant superconducting surface code. Gate errors could besuppressed further by a simple circuit redesign to increase qubitanharmonicity or by efforts to improve the timing stability of the SFQcircuit. Herein, only simple SFQ pulse trains have been considered,although other pulse sequences are possible. Specifically,state-of-the-art SFQ timing generators should allow the realization ofrobust sequences with arbitrary inter-pulse delays. In addition, it isanticipated that optimal control tools such as those used to optimizemicrowave-based single qubit gates and fast two-qubit gates could alsobe employed to modify SFQ sequences with controlled inter-pulse delaysdesigned to suppress |2

state errors and increase gate speed and/or fidelity over the naive gatesequences considered here.

Due to technical complexities of transmitting SFQ pulses from chip tochip, the practical realization of SFQ-based qubit gates may require theon-chip integration of the qubit circuit with at least a handful of SFQelements. While in the past the high static dissipation of SFQ circuitshas presented an obstacle to milliKelvin-scale temperature operation,the recent development of low-power biasing schemes for reciprocalquantum logic (RQL) and energy efficient SFQ logic (eSFQ) opens the doorto the integration of SFQ and qubit circuits on the same chip. Care mustbe taken to isolate the qubit circuit from nonequilibrium quasiparticlesgenerated in the SFQ control circuit; however, quasiparticle poisoningof the qubit circuit can be mitigated by avoiding direct galvanicconnection between the signal and ground traces of the SFQ and qubitcircuits. The ability to generate fluxons in close proximity to thequbit circuit may provide a high degree of robustness to the SFQ-basedrotations, due to the quantization of flux associated with the SFQpulses.

In conclusion, the present disclosure provides a system and methods forthe high-fidelity coherent manipulation of quantum systems modes usingresonant trains of SFQ pulses. The SFQ pulse trains can be generatedlocally in a qubit cryostat without need for external microwavegenerators. Taken together with a recent proposal to map the quantuminformation in a cQED circuit to a binary digital output using Josephsonmicrowave photon counter, the approach of the present disclosure pointsin a direction toward the integration of large scale superconductingquantum circuits with cold control and measurement circuitry based onSFQ digital logic.

The present invention has been described in terms of one or morepreferred embodiments, and it should be appreciated that manyequivalents, alternatives, variations, and modifications, aside fromthose expressly stated, are possible and within the scope of theinvention.

The invention claimed is:
 1. A quantum computing system comprising: atleast one superconducting quantum circuit described by multiple quantumstates; at least one single flux quantum (“SFQ”) control circuitconfigured to generate a voltage pulse sequence that includes aplurality of voltage pulses temporally separated by a pulse-to-pulsespacing timed to a resonance period; and at least one coupling extendingbetween the at least one superconducting quantum circuit and the atleast one SFQ control circuit and configured to transmit the voltagepulse sequence generated using the SFQ control circuit to the at leastone superconducting quantum circuit.
 2. The system of claim 1, whereinthe at least one superconducting quantum circuit includes at least aqubit circuit.
 3. The system of claim 1, wherein the at least onesuperconducting quantum circuit includes at least a resonant cavitycircuit.
 4. The system of claim 1, wherein the resonance period isconsistent with a target transition between the multiple quantum states.5. The system of claim 1 further comprising a controller systemconfigured to optimize the pulse-to-pulse spacing to minimize a gateinfidelity due to at least one of a timing error, a timing jitter and aweak qubit anharmonicity.
 6. The system of claim 1, wherein the couplingincludes at least one of a capacitive coupling and an inductivecoupling.
 7. The system of claim 1, wherein the coupling includes aJosephson transmission line consisting of multiple Josephson junctions.8. A method for controlling superconducting quantum circuits, the methodcomprising: providing at least one superconducting quantum circuitdescribed by multiple quantum states and coupled to at least one singleflux quantum (“SFQ”) control circuit; generating, using the at least oneSFQ control circuit, a voltage pulse sequence that includes a pluralityof voltage pulses temporally separated by a pulse-to-pulse spacing; andapplying the voltage pulse sequence to the at least one superconductingquantum circuit to achieve an excitation consistent with a targettransition between the multiple quantum states.
 9. The method of claim8, wherein the at least one superconducting quantum circuit includes atleast a qubit circuit.
 10. The method of claim 9, wherein applying thevoltage pulse sequence to the qubit circuit generates discrete rotationsinterspersed with free precession intervals timed to a qubit period. 11.The method of claim 8, wherein the at least one superconducting quantumcircuit includes at least a resonant cavity circuit.
 12. The method ofclaim 8, wherein the pulse-to-pulse sequence is timed to a resonanceperiod of the at least one superconducting quantum circuit.
 13. Themethod of claim 12, wherein the resonance period is consistent with thetarget transition.
 14. The method of claim 8 further comprisingoptimizing the pulse-to-pulse spacing to minimize a gate infidelity dueto at least one of a timing error, a timing jitter and a weak qubitanharmonicity.
 15. A method for controlling superconducting quantumcircuits, the method comprising: providing at least one superconductingquantum circuit described by multiple quantum states and coupled to atleast one single flux quantum (“SFQ”) circuit; generating, using the atleast one SFQ control circuit, a voltage pulse sequence that includes aplurality of voltage pulses temporally separated by a pulse-to-pulsespacing timed to a resonance period; and applying the voltage pulsesequence to the at least one superconducting quantum circuit to achievean excitation consistent with a target transition between the multiplequantum states.
 16. The method of claim 15, wherein the at least onesuperconducting quantum circuit includes at least a qubit circuit. 17.The method of claim 16, wherein applying the voltage pulse sequence tothe qubit circuit generates discrete rotations interspersed with freeprecession intervals timed to a qubit period.
 18. The method of claim15, wherein the at least one superconducting quantum circuit includes atleast a resonant cavity circuit.
 19. The method of claim 15, wherein theresonance period is consistent with the target transition.
 20. Themethod of claim 15 further comprising optimizing the pulse-to-pulsespacing to minimize a gate infidelity due to at least one of a timingerror, a timing jitter and a weak qubit anharmonicity.